What do you think are the most essential processes and functions for a successful ITSM program?

The answer to ANY "how-to" or "most important" question regarding ITIL and ITSM must begin with "It depends..." There isn't a universal; one size fits all response.

ITSM is about SERVICE management, not processes. Processes (and functions) are a means to an end -- allowing the IT service provider to mature and develop the necessary organizational skills to provide value to the business processes that itsm and agile serves. The most critical methods to address immediately are the ones that get in the way.

With respect to functions, I don't know how to answer this question because the tasks described in the ITIL Service Operation book aren't intended to be org-chart entities, they are abstractions for how the organizations align people into teams with appropriate tools and governance. Again, the answer has to start with, "It depends..." In a small organization, a handful of people might have to do everything. In larger organizations, there's more specialization. In either case (and everything in between) the resulting org-chart might not have anything that looks like anything described in the ITIL books.

The most critical processes and functions are the ones that contribute to immediate IT service provider capability to meet (and eventually exceed) business need.

The question about how to go about addressing adoption, improvement, tweaking (etc.) processes and functions to meet the need is separate (that again begins with, "It depends..."